A102781 - OEIS

%I #18 Oct 14 2024 00:10:27

%S 0,1,2,3,5,6,8,9,11,14,15,18,20,21,23,26,29,30,33,35,36,39,41,44,48,

%T 50,51,53,54,56,63,65,68,69,74,75,78,81,83,86,89,90,95,96,98,99,105,

%U 111,113,114,116,119,120,125,128,131,134,135,138,140,141,146,153,155,156

%N Number of positive even numbers less than the n-th prime.

%C Same as A005097 ((odd primes - 1)/2) with a leading zero. - Lambert Klasen, Nov 06 2005

%H Ray Chandler, <a href="/A102781/b102781.txt">Table of n, a(n) for n = 1..10000</a>

%F Integer part of p#/((p-2)#*2#), where p=prime(n) and i# is the primorial function A034386(i). - _Cino Hilliard_, Feb 25 2005

%F n# = product of primes <= n. 0# = 1# = 2. [This is not a standard convention!] n#/(n-r)#/r# is analogous to the number of binomial coefficients A007318 = C(n, r) = n!/(n-r)!/r! where factorial ! is replaced by primorial #.

%F a(n) = prime_n - floor((prime_n)/2) - 1. - _Giovanni Teofilatto_, Nov 05 2005

%F a(n) = [A034386(prime(n))/(2*A034386(prime(n)-2))], n>2. - _R. J. Mathar_, May 18 2009

%F a(n) = [(prime(n)-1)/2] where the integer part [.] needs be taken only for n=1. - _M. F. Hasler_, Dec 13 2019

%t Table[Prime[n] - Floor[Prime[n]/2] - 1, {n, 65}] (* _Robert G. Wilson v_ *)

%o (PARI) c(n,r) = { local(p); forprime(p=r,n, print1(floor(primorial(p)/ primorial(p-r)/primorial(r)+.0)",") ) } primorial(n) = \ The product primes <= n using the pari primelimit. { local(p1,x); if(n==0||n==1, return(2)); p1=1; forprime(x=2,n,p1*=x); return(p1) }

%o (PARI) apply( A102781(n)=(prime(n)-1)\2, [1..99]) \\ _M. F. Hasler_, Dec 13 2019

%o (Python)

%o from sympy import prime

%o def A102781(n): return prime(n)-1>>1 # _Chai Wah Wu_, Oct 13 2024

%Y Cf. A005097. - _R. J. Mathar_, May 18 2009

%Y Equals (A000040-1)/2, integer part (0) for the first term. - _M. F. Hasler_, Dec 13 2019

%K easy,nonn

%O 1,3

%A _Cino Hilliard_, Feb 25 2005

%E Simpler definition from _Giovanni Teofilatto_, Nov 05 2005

%E Edited by _N. J. A. Sloane_ Jul 05 2009 at the suggestion of R. J. Mathar